underactuated wheeled inverted pendulum vehicle based on a data-driven trajectory planner

Accepted Manuscript Indirect adaptive fuzzy control for a nonholonomic/underactuated wheeled inverted pendulum vehicle based on a data-driven trajectory planner

Ming Yue, Cong An, Yu Du, Jianzhong Sun

PII: DOI: Reference:

S0165-0114(15)00371-1 http://dx.doi.org/10.1016/j.fss.2015.08.013 FSS 6878

To appear in:

Fuzzy Sets and Systems

Received date: Revised date: Accepted date:

19 October 2014 25 July 2015 17 August 2015

Please cite this article in press as: M. Yue et al., Indirect adaptive fuzzy control for a nonholonomic/underactuated wheeled inverted pendulum vehicle based on a data-driven trajectory planner, Fuzzy Sets and Systems (2015), http://dx.doi.org/10.1016/j.fss.2015.08.013

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Fuzzy Sets and Systems 00 (2015) 1–17

Fuzzy Sets and Systems

Indirect adaptive fuzzy control for a nonholonomic/underactuated wheeled inverted pendulum vehicle based on a data-driven trajectory planner Ming Yuea,b , Cong Ana , Yu Dua , and Jianzhong Sunc a School

of Automotive Engineering, Dalian University of Technology, Dalian, Liaoning Province, 116024 China; e-mail: [email protected] b State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, Heilongjiang Province, 150080 China c Department of Electrical and Electronics Engineering, Dalian University of Technology, Dalian, Liaoning Province, 116024 China

Abstract In this study, we investigate an error data-based trajectory planner and indirect adaptive fuzzy control for a class of wheeled inverted pendulum vehicle systems. Based on the error dynamics, the closed-loop trajectory planner can generate the desired velocity values. Using the virtual acceleration input for the tilt angle subsystem, composite control for the rotational and longitudinal subsystems can be constructed via indirect adaptive fuzzy and sliding mode control approaches to achieve simultaneous velocity tracking and tilt angle stabilization. We rigorously prove the system stability and convergence of the tracking error signals using the Lyapunov theory and LaSalle’s invariance theorem. The results of our numerical simulations demonstrated the efficiency of the proposed control strategies and the implementations of the algorithms. Keywords: adaptive mechanism, data-driven, indirect fuzzy control, nonholonomic/underactuated, wheeled inverted pendulum.

1. Introduction A wheeled inverted pendulum (WIP) vehicle possesses only two active driving wheels but it can implement nearly every mode of motion, such as advancing, steering, and stopping. Over the past few years, there has been extensive interest in the development of these types of vehicles (e.g., [1, 2, 3, 4]). Due to their advantageous features, such as compact construction, convenient operation, high maneuverability, and low fuel consumption, WIP vehicles can be employed in emerging applications in commercial, civilian, and military areas. However, the efficient control of the WIP platform is a challenging problem, mainly because: (i) it is difficult to use current modeling techniques to obtain an accurate vehicle dynamic model, which is generally highly nonlinear, timevarying, and coupled in nature; (ii) the system suffers from a nonholonomic constraint on the kinematic level, which traditional control methods are unable to use directly; and (iii) the dynamics of the suspension possess obvious underactuated characteristics, which means that the number of control inputs is less than the number of degree of freedom. All of these difficult issues are hindrances to the control community [5, 6, 7]. Many previous studies have focused on control methods to implement trajectory tracking for WIP vehicles, such as conventional proportional-integral-derivative (PID) control, feedback stabilization, sliding mode schemes, and intelligent control. It is well known that PID control is a simple and easy approach but due to the complexities of WIP vehicles, it is a difficult to select three appropriate gains [8, 9]. Thus, to improve the system control performance, feedback stabilization based on an approximate linear model is widely employed. For example, in [10], a two-level controller was proposed for tracking and stabilizing the vehicle’s posture based on a partial linear model where the internal dynamics were isolated. Similar methods were also employed by [11, 12]. In addition, the variable structure method is another powerful approach for controlling WIP vehicles. For instance, Yue et al. employed a sliding mode scheme to effectively control a two-wheeled vehicle with a dropping pendulum-like suspension, where the mobile platform was fairly similar to that in a WIP vehicle [13]. However, most of these controllers rely on the system model information, which can rarely be obtained in practice. It should also be mentioned that many advanced modeling and identification techniques can be used for model-based control design (e.g., [14, 15, 16]), but it is still difficult to obtain an explicit description of the vehicle in real time. Intelligent 1

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control, including the use of fuzzy control and neural networks (NNs), has been shown to be a powerful technique for control design when dealing with complex dynamical systems (e.g., [17, 18, 19, 20]). In particular, Li et al. developed an output feedback adaptive NN controller with a linear dynamic compensator to achieve stable dynamic balance of the vehicle body, while tracking of the desired trajectories could also be achieved at the same time [21, 22]. Moreover, in [23], an adaptive fuzzy logic control system was explored based on updated laws for a WIP vehicle and the control performance was improved. Based on these studies, it is clear that intelligent control is an effective tool for controlling WIP vehicles, which have complex dynamics and nonholonomic/underactuated behaviors. Furthermore, compared with the aforementioned control approaches, fuzzy control can usually achieve better control performance because it does not depend on an exact mathematical model and it can effectively synthesize the successful experiences of operators using linguistic instructions [24, 25, 26, 27, 28]. In general, fuzzy control comprises indirect fuzzy control and direct fuzzy control. In contrast to direct fuzzy control, which is based on the difference between the performance of the actual system and the ideal performance level, indirect fuzzy control applies the universal approximation theorem to obtain system models of the controlled object, which is appropriate for merging the other control techniques that comprise a composite controller in an entire underactuated system [29, 30]. Surprisingly, direct fuzzy control has been used widely to control WIP vehicles (e.g., [31, 32, 33]), but few studies have employed the indirect fuzzy scheme to control this type of wheeled mobile platform. In addition to the underactuated behaviors of WIP vehicles, the nonholonomic constraint is another unavoidable problem when a vehicle tracks the reference trajectories. If the wheeled mobile platform moves while the wheelterrain satisfies the pure rolling condition, then the constraint cannot be integrated into a linear form [34, 35]. According to Brockett’s necessary condition for asymptotic stabilization, no continuous time-invariant state-feedback controller exists that can asymptotically stabilize the mobile platform [36]. For a WIP vehicle, the combination of underactuated suspension and the nonholonomic constraint represent a challenging task during control system synthesis. With the exception of a few previous studies, the tracking objective has focused on tracking the longitudinal and rotational velocities, whereas tracking of the position trajectory was described in the Cartesian frame. Therefore, simultaneously addressing the underactuated and nonholonomic problems for WIP vehicles is still an open problem, and thus the motivation of the present study. In the present study, we propose a novel data-driven closed-loop trajectory planner that differs from the traditional path planning, which solves the nonholonomic constraint and underactuated control issues on kinematic and dynamic levels, respectively. With respect to the trajectory planner, we conceive a hypothesized vehicle with the desired posture values and we then establish the error dynamics based on the posture tracking error data so we can apply Lyapunov stability theory to guarantee the stability of the closed-loop planner. It should be noted that the closed-loop trajectory planner is similar to a close-loop system based on the vehicle kinematics, where the planner’s outputs are regarded as the desired longitudinal and rotational velocities for the vehicle’s dynamic system. This treatment allows the data-driven trajectory planner to generate the desired values in real time, where the closed-loop feedback characteristics can enhance the robustness of the planner system against unavoidable parameter variation and unpredictable external disturbances. In addition, we propose composite control by introducing a virtual control method to address the underactuated problem. To explore the underactuated behaviors, the overall system is divided into three subsystems, i.e., rotational velocity, longitudinal velocity, and tilt angle subsystems, where the latter two subsystems are considered together because they are controlled by only one input. In our further analysis, the acceleration of the mid-point of the vehicle axis is assumed to be a virtual control input that allows the transformation of the underactuated behaviors into actuated behaviors. Therefore, we present a composite controller that combines indirect fuzzy control and the variable structure technique, which can guarantee the simultaneous tracking of the desired longitudinal velocity as well as stabilizing the tilt angle of the suspension. In summary, the main contributions of the present study are as follows. 1) A dynamic model is proposed via the Lagrangian methodology, which can clearly describe the coupling effects, time-varying behaviors, and nonlinear characteristics of WIP vehicles. 2) Based on the tracking errors, a closed-loop trajectory planner is presented, which can resolve the nonholonomic constraint and underactuated problems on the kinematic and dynamic levels, respectively. 3) A composite controller is constructed according to indirect fuzzy control and the variable structure technique, where the concept of virtual acceleration input is proposed to allow the simultaneous tracking of velocities and stabilization of the tilt angle. The remainder of this paper is organized as follows. In Section 2, we introduce the mathematic model of a WIP vehicle and we state the problem formulation. In Section 3, we present the error data-driven theory and we describe the closed-loop trajectory planner system. The indirect adaptive fuzzy control method is explained in Section 4 and the procedures involved in the design process are also described in detail. In Section 5, we consider the specific design methods for the three main subsystems and the results of numerical simulations are presented that verify the effectiveness of the proposed control strategy. In Section 7, we give our conclusions. 2

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2. Problem formulation and preliminaries In this section, we describe the kinematics and dynamics of a WIP vehicle, as well as presenting some preliminary details and formulations. The vehicle parameters and variables used to describe the vehicle system are shown in Fig.1 and Table 1.

Figure 1. Schematic diagram showing the vehicle’s characteristic parameters.

Table 1. Key parameters related to a WIP vehicle

Notation x, y θl , θr θ ϕ υ ω mp mc mw Jv Jc Jw r l g d

Definition Current position of the vehicle on the XOY plane Rotational angles of the left and right wheels Tilt angle of the vehicle body Rotational angle of the vehicle Longitudinal velocity of the vehicle Rotational velocity of the vehicle Mass of the inverted pendulum Mass of the chassis Mass of each wheel Moment of inertia for the chassis and pendulum about the Z-axis Moment of inertia for the chassis about the Y-axis Moment of inertia for the wheel with respect to the Y-axis Radius of the wheels Distance from the body center of gravity to the wheel axis Acceleration due to gravity Distance between the two wheels along the axle center

2.1. General underactuated system The dynamics of the WIP vehicle with an underactuated body subject to a nonholonomic constraint can be formulated by M(q)q¨ + Vm (q, q) ˙ q˙ + G(q) = B(q)τ + AT λ, (1) where q = [q1 q2 q3 q4 ]T = [x y ϕ θ]T ∈ R4 is the vector of the generalized coordinates, the elements of which are ˙ q˙ ∈ R4 is the matrix of the Coriolis and centrifugal independent, M(q) ∈ R4×4 denotes the inertia matrix, Vm (q, q) forces, G(q) ∈ R4 is the vector of gravitational forces, B(q) ∈ R4×2 is the matrix relative to the control efforts, τ ∈ R2 is the vector of control inputs, AT λ ∈ R4 is the vector of constraint forces, where AT ∈ R4 is a Jacobian matrix, and λ ∈ R comprise Lagrangian multipliers that correspond to the system constraints. 3

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Furthermore, if this is an incomplete controllable system, then the entire system can be divided into two parts: actuated and underactuated subsystems. We define q = [qa qu ]T , where qa and qu denote the actuated and underactuated state variables, and the system (1) can be rewritten as           τ Vau V G B Bu Ma Mau ; τ= a , ˙ = a ; Vm (q, q) ; G(q) = a ; B(q) = a M(q) = Mua Mu Vua Vu Gu 0 0 0 where Ma and Mu are the inertia submatrices relative to the actuated and underactuated parts, respectively. Mau and Mua are the coupling inertia matrices of the two subsystems. Similarly, Va and Vu denote the centripetal and Coriolis submatrices, and Vau and Vua are related to their coupling effects. Ga and Gu are the corresponding gravitational torque terms for the actuated and underactuated subsystems, respectively. τa is the control input vector for the actuated subsystem [37, 38]. 2.2. Reduced dynamics In terms of the actuated subsystem, the system constraints, including the nonholonomic constraints, can be described by S aT (q)q˙ a = 0, (2) where S a is the kinematic constraint matrix related to the system constraints. The effect of the constraints can be taken as a restriction of the dynamics on the manifold Ωn as follows. 

Ωn = {(qa , q˙ a )|S aT q˙ a = 0}

(3)

Assume that the annihilator of the co-distribution spanned by the covector fields S a is a smooth nonsingular distribution Δ. This distribution Δ is spanned by a set of smooth and linearly independent vector fields A1 (q) and  A2 (q), i.e., Δ = span{A1 (q), A2 (q)}, for which the local coordinates satisfy the following relation: S aT AT = 0,

(4)



where A = [A1 (q), A2 (q)], and we note that AT A is of full rank. Constraint (2) implies the existence of the vector ν such that q˙ a = S a (q)ν. (5) Moreover, let z = [ν q˙ u ]T be a new coordinate of the system. By differentiating (5) with respect to time, substituting the results into the system (1), and multiplying S (q) = diag[S aT I] on both sides of the equation to eliminate the kinematic nonholonomic constraint AT λ, we obtain the reduced order dynamics of the WIP vehicle as follows: ¯ = B(z)τ, ¯ ¯ z + V¯ m (z, z˙)z + G(z) (6) M(z)˙ where

 T S M S ¯ M(z) = a a a Mua S a

  T S aT Mau S M S˙ + S aT Va S a ; V¯ m (z, z˙) = a a ˙ a Mu Mua S a + Vua S a  T   T  S B S a Ga ¯ ¯ ; B(z) = a a . G(z) = Gu 0

 S aT Vau ; Vu

2.3. Dynamics of the WIP vehicle The facts that the stated variables for the WIP vehicle are coupled together and they exhibit strong nonlinear characteristics make it difficult to establish a dynamic model of this type on a mobile platform. In this subsection, using the Lagrangian-Eulerian method, we derive a dynamic model for to allow the synthesis of the system. Assumption 1. Sufficient friction between the wheels of the mobile platform and the ground can be guaranteed such that the assumption of nonholonomic constraints holds. Define q = [qa qu ]T , where qa = [x y ϕ]T and qu = θ. If the pure rolling constraint is satisfied, then the υ ωd υ ωd velocities of the left and right wheels are governed by θ˙r = + and θ˙l = − . The system energy can be r 2r r 2r divided into three parts: the kinetic energies for the wheels T w , the vehicle body T c , and the pendulum T p , which

4

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are calculated, respectively, by Tw

=

Tc

=

Tp

=

 υ 2 1 1 1 Jw  + mw υ2 + mw + 2 d2 ω2 Jw (θ˙l2 + θ˙r2 ) + mw (υ2l + υ2r ) = Jw 2 2 r 4 r 1 1 ˙2 1 2 2 mc υ + Jc θ + Jv ϕ˙ 2 2 2 1 1 1 2 m p (υ + lθ˙ cos θ) + m p (−lθ˙ sin θ)2 + m p (l sin θ)2 ω2 . 2 2 2

(7) (8) (9)

In addition, the total potential energy is U = −m p gl(1 − cos θ).

(10)

Therefore, the Lagrangian multiplier can be obtained by L = Tw + Tc + T p − U =

  1 2Jw 1 ˙2 1 Jc θ + m p ( x˙ cos ϕ + y˙ sin ϕ + lθ˙ cos θ)2 + mc + 2mw + 2 ( x˙ cos ϕ + y˙ sin ϕ)2 2 2 2 r 1 Jw  2 2 1 1 + mw + 2 d ϕ˙ + m p l2 (θ˙ 2 + ϕ˙ 2 ) sin2 θ + Jv ϕ˙ 2 + m p gl(1 − cos θ). 4 2 2 r

(11)

Using the generalized coordinates as q = [q1 q2 q3 q4 ]T = [x y ϕ θ]T , a dynamic model of the WIP vehicle can be obtained using Lagrange’s dynamical equation as follows:   ∂L d ∂L = ui , (i = 1, . . . , 4), (12) − dt ∂q˙ i ∂qi where ui represents the generalized force with respect to the corresponding general coordinate qi . Furthermore, ˙T system (1) can be transformed into the reduced dynamics (6), where the transfer matrix between q and z = [ω υ θ] can be formulated as follows. ⎡ ⎤  ⎢⎢⎢⎢cos ϕ 0 0⎥⎥⎥⎥  ⎢⎢ sin ϕ 0 0⎥⎥⎥ S (q) 0 ⎥ = ⎢⎢⎢⎢ (13) S (q) = a 0 I 1 0⎥⎥⎥⎥⎦ ⎢⎢⎣ 0 0 0 1 Therefore, the nonholonomic constraint can be eliminated from the dynamic model (1) since S T AT = 0. Using the aforementioned computational procedures, a reduced order dynamics with the same form as (6) can ultimately be obtained, where the specific matrices are as follows: ⎤ ⎡ ⎤ ⎡   0 m p l cos θ ⎥⎥⎥ ⎢⎢⎢ 1 1⎥⎥⎥ ⎢⎢⎢ mv 1 τl ⎥ ⎢ ⎥⎥⎥ ⎢ 2 2 ⎥ ⎢ ⎢ ¯ ¯ ⎥ ⎢ ⎢ M(q) = ⎢⎢ 0 Jϕ + m p l sin θ 0 ⎥⎥⎦ ; B = ⎢⎢⎣−d d⎥⎥⎦ ; ua = τ ; ⎣ r r 0 0 m p l cos θ 0 Jc + m p l2 ⎡ ⎢⎢⎢0 ⎢⎢⎢⎢ ˙ = ⎢⎢⎢⎢0 V¯ m (q, q) ⎢⎢⎢ ⎣0

0 1 − m p l2 θ˙ sin 2θ 2 1 − m p l2 ϕ˙ sin 2θ 2

⎤ −m p lθ˙ sin θ ⎥⎥ ⎡ ⎤ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ 0 1 ⎥ 2 ⎥ ⎢ ⎥⎥⎥ − m p l ϕ˙ sin 2θ⎥⎥⎥ ; G(q) = ⎢⎢⎢ 0 ⎥⎥⎦ , 2 ⎥⎥⎥ ⎢⎣ ⎥⎦ −m p gl sin θ 0

1 Jw 2Jw where Jϕ = Jv + (mw + 2 )d2 and mv = m p + mc + 2mw + 2 , both of which are introduced for simplicity. It is 2 r r interesting to note that the nonholonomic constraint has been eliminated from the system. ¯ is symmetric and positive definite. Property 1. The inertia matrix M ¯˙ − 2V¯ m is skew-symmetric. Property 2. The matrix M 2.4. Three subsystems Note that the longitudinal and rotational movements of the vehicle are driven by the summation and difference of the left and right wheel torques. Hence, we define the new control inputs as follows. τυ

= τl + τ r

(14)

τω

= −τl + τr

(15)

5

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According to (6), the vehicle dynamics are related to four variables: the vehicle rotational velocity ω, longitudinal ˙ For the purposes of control, the reduced order system (6) velocity υ, the tilt angular θ, and its angular velocity θ. can be partitioned into three subsystems as follows: 1) ω-subsystem: ˙ ϕ) ˙ + gω (θ) ω ˙ = fω (θ, θ, m p l2 ϕ˙ θ˙ sin 2θ d + τω , =− Jϕ + m p l2 sin2 θ r(Jϕ + m p l2 sin2 θ)

(16)

˙ ϕ) ˙ + gυ (θ) υ˙ = fυ (θ, θ,  Jθ 1 = Jθ m p lθ˙ 2 sin θ − m2p l3 ϕ˙ 2 sin θ cos2 θ − m2p l2 g sin θ cos θ + τυ , Ω Ωr

(17)

˙ ϕ) ˙ + gθ (θ) θ¨ = fθ (θ, θ,   m p l cos θ 1 = 2mv m p gl sin θ + mv m p l2 ϕ˙ 2 sin 2θ − m2p l2 θ˙ 2 sin 2θ − τυ , 2Ω Ωr

(18)

2) υ-subsystem:

3) θ-subsystem:

where Jθ = Jc + m p l2 and Ω = mv Jθ − m2p l2 cos2 θ. Thus, we conclude that Jϕ + m p l2 sin2 θ > 0, and Ω = mv Jθ − m2p l2 cos2 θ is also positive for all θ ∈ R. Therefore, for a practical mechanical system, the three subsystems (16–18) have no singular point globally. 3. Dynamic trajectory planning 3.1. Error data-driven planner The three subsystems are established essentially on the level of system dynamics, but in practice, the desired trajectories are usually given on Cartesian coordinates. Therefore, it is always a requirement that in addition to holding the vehicle body around the upright position, the actual configuration center of the WIP vehicle described by x, y, and ϕ should converge rapidly to the reference trajectory with the desired values of xd , yd , and ϕd , respectively. Thus, the trajectory tracking control objective for the WIP vehicle with the reduced order system can be summarized as: find the control inputs τω and τυ such that ω → ωd and υ → υd as t → ∞, as well as the vehicle body that holds an upright position with the unstable equilibrium, i.e., θ → 0 as t → ∞. Assumption 2. All of the desired trajectories, including ωd and υd , and their time derivatives up to the third order are continuously differentiable and bounded for all t ≥ 0. For the desired tilt angle of the WIP vehicle, it is expected that θd = 0 and θ˙d = 0. We propose a trajectory planner, which can be used to generate the desired signals for the inner dynamic systems, while the posture errors will converge to zero within a finite time. It should be noted that from another perspective, this planner can be taken as a closed-loop control system. The trajectory planner is illustrated in Fig.2. Let q(t) ¯ = [x y ϕ]T be a vector in the set of the actual vehicle’s posture information and define q¯ d (t) = T [xd yd ϕd ] as the vector describing a hypothesized vehicle that needs to be tracked, then a tracking error space q¯ e = q¯ d − q¯ = [xe ye ϕe ]T is deduced in the following manner. It is interesting to note that from the viewpoint of control, an appropriate input νc = [υc ωc ]T should be found to achieve the objective lim q¯ e (t) = 0, but from the t→∞ planner’s viewpoint, the planner should generate νc as the desired trajectories for the reduced order subsystems (16)–(18). By further calculation and analysis, the posture errors can be obtained as follows. ⎤⎡ ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ xe ⎥⎥⎥ ⎢⎢⎢ cos ϕ sin ϕ 0⎥⎥⎥ ⎢⎢⎢ xd − x ⎥⎥⎥ ⎥⎢ ⎢ ⎥ ⎢ ⎥ (19) q¯ e = ⎢⎢⎢⎢ ye ⎥⎥⎥⎥ = ⎢⎢⎢⎢− sin ϕ cos ϕ 0⎥⎥⎥⎥ ⎢⎢⎢⎢ yd − y ⎥⎥⎥⎥ ⎦⎣ ⎣ ⎦ ⎣ ⎦ 0 0 1 ϕd − ϕ ϕe By differentiating (19) and considering the nonholonomic constraint (i.e., x˙ sin ϕ = y˙ cos ϕ), the posture error dynamics can be achieved as follows. ⎡ ⎤ ⎡ ⎤ ⎢⎢⎢ x˙e ⎥⎥⎥ ⎢⎢⎢υd cos ϕe + ωye − υ⎥⎥⎥ ⎥ ⎢ ⎢ ⎥ (20) q¯˙ e = ⎢⎢⎢⎢ y˙ e ⎥⎥⎥⎥ = ⎢⎢⎢⎢ υd sin ϕe − ωxe ⎥⎥⎥⎥ ⎣ ⎦ ⎣ ⎦ ϕ˙ e ωd − ω 6

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xd yd φd

ex ey eφ x y φ

Trajectory Planer Module ωc Planner

υc

eω eυ θ

Sensors

Indirect adaptive fuzzy controllers ω-subsystem Controller

7

WIP vehicle

τω

υ-subsystem uυ + τυ controller + uθ θ-subsystem controller

ω-subsystem

υ-subsystem θ-subsystem

ω υ θ

Figure 2. Schematic diagram of the indirect adaptive fuzzy control system.

Remark 1. Note that ωc and υc are different from the desired values of ωd and υd . In practice, the desired trajectory for an actual vehicle is always given by the vehicle posture variables xd , yd and ϕd , which can further compute the desire velocities ωd and υd in advance, but this transformation is irreversible. The trajectory planner will generate ωc and υc to provide a connection between the reduced order subsystems and vehicle postures system. 3.2. Implementation of the closed-loop planner If the trajectory planner employs a closed-loop structure, then it will be highly robust to overcome the initial bias deviation as well as rejecting the outside disturbances. Based various studies, a controller derived by the direct Lyapunov methodology (e.g., [34, 35]) can be employed as a planner for the WIP vehicle, as follows:     υc υd cos ϕe + λ3 xe = , (21) ωc ωd + λ1 υd ye + λ2 sin ϕe where λ1 , λ2 , and λ3 are positive constants. Then, a theorem related to the trajectory planner can be summarized as follows. Theorem 3.1. As an error data-based driven system, (19) and (20), then by using the trajectory planner (21), all of the signals in the closed-loop planner system are bounded and the tracking errors will converge asymptotically to zero, i.e.,   lim |xe (t)| + |ye (t)| + |ϕe (t)| = 0. t→∞

Proof. Define a candidate Lyapunov function as follows. V1 =

1 2 1 2 1 x + y + (1 − cos ϕe ) 2 e 2 e λ1

(22)

Clearly, V1 ≥ 0. Differentiating V1 with respect to time yields 1 V˙ 1 = xe x˙e + ye y˙ e + ϕ˙ e sin ϕe λ1 = xe (υd cos ϕe + ωye − υ) + ye (υd sin ϕe + ωxe ) +

1 (ωd − ω) sin ϕe λ1

.

(23)

After substituting (21) into the equation above, it holds that λ2 sin2 ϕe V˙ 1 = −λ3 xe2 − λ1

(24)

. Obviously, V˙ 1 ≤ 0. According to the Lyapunov stability criterion and the LaSalle invariance principle, for all initial conditions q(0), ¯ the corresponding solution qe converges to zero. Thus, the theorem is proved. 4. Indirect adaptive fuzzy control Fuzzy control is used widely for sophisticated nonlinear systems because it is not dependent on an accurate model of the control object. In addition, the fuzzy logic can merge the experiences of experts or operators in 7

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the system controller, thereby making the closed-loop system much more intelligent and flexible under uncertain disturbances. Fuzzy control can be divided into direct and indirect control methods. In terms of the adaptability, indirect fuzzy control is better than the direct method because the system model information utilized in the controller can be updated online, thereby reducing the dependency on accurate mechanical parameters [39, 40]. 4.1. Problem description Consider a nonlinear system with n-order as follows:     x¯(n) = f x¯, x˙¯, · · · , x¯(n−1) + g x¯, x˙¯, · · · , x¯(n−1) u

(25)

, where f (·) and g(·) are unknown nonlinear functions, and u ∈ Rn is the control input. Suppose that the desired value is y¯ m . Then, the tracking error can be defined as e = y¯ m − y¯ = y¯ m − x¯ and it has e = [e e˙ , · · · , e(n−1) ]T . If the feedback gains are set as k = [k¯ n , · · · , k¯ 1 ]T , where k¯ i is the coefficient, which can make all the roots of the polynomial sn + k¯ 1 sn−1 + · · · + k¯ n on the left half of the complex plane, then the controller can be designed as follows.  1  T u∗ = − f (x) + y¯ (n) (26) m + k e g(x) By substituting (26) into (25), we have e(n) + k¯ 1 e(n−1) + · · · + k¯ n e = 0.

(27)

Considering the characteristic of k, this means that e(t) → 0 as t → ∞, i.e., the system output y¯ can converge asymptotically to its desired output y¯ m . 4.2. Fuzzy logic mechanism In practice, it is almost impossible to obtain accurate values of f (·) and g(·) in advance; therefore, the fuzzy logic mechanism can be employed to estimate the true values of f (·) and g(·) online based on their corresponding updated values fˆ(·) and gˆ (·). By the universal approximate theorem of the fuzzy logic mechanism, the estimated errors can be controlled to the desired boundedness as much as possible. Taking fˆ(x|Θ) as an example, the estimated value can be formulated according to three main steps, as follows. Step 1 For the variables x¯i (i = 1, 2, · · · , n), define the fuzzy set Alii (li = 1, 2, · · · , pi ), where pi is the number of the fuzzy set.  Step 2 Construct the fuzzy logic system fˆ(x|Θ) using the fuzzy rules with the number of pi , as follows: n

i=1

R( j) : IF x¯1 is Al11 and · · · and x¯n is Al1n THEN fˆ is El1 ···ln ,

(28)

where li = 1, 2, · · · , pi and i = 1, 2, · · · , n. Step 3 The outputs of this fuzzy logic system with the center-average defuzzifier, product inference, and singleton fuzzier are of the following form:   p1 pn n    ··· y¯ lf1 ···ln μAli ( x¯i ) i i=1 l1 =1 ln =1  ,  (29) fˆ(x|Θ f ) = p1 pn  n   ··· μAli ( x¯i ) l1 =1

ln =1 i=1

i

where μA j ( x¯i ) is the membership function for x¯i . Let y¯ lf1 ···ln be the free parameter and collect them in the set i

n 

as Θ f ∈ R

i=1

pi

. Define vector the ξ(x), and then (29) can be replaced by fˆ(x|Θ f ) = ΘTf ξ(x),

(30)

where ξ(x) is a vector defined as n 

ξl1 ,··· ,ln (x) =

p1  l1 =1

8

μAli ( x¯i ) i .  n  μAli ( x¯i )

i=1 pn 

···

ln =1 i=1

i

(31)

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9

4.3. Indirect adaptive fuzzy control By selecting reasonable design parameters, the controllers with the estimated values can approach the appropriate controllers with accurate values to any arbitrary tolerance. The indirect fuzzy controller (Fig.3) can be outlined as:

­ x(n) ® ¯y x

Plant f ( x )  g ( x )u

ym

-+

Indirect adaptive fuzzy controller 1 [ fˆ ( x 4 f )  ym( n )  k T e] gˆ( x 4 g )

u

Updated laws

4 f (0), 4 g (0)

4f

b1;[ ( x)

4g

b2;K( x)u

Figure 3. Schematic diagram of indirect adaptive fuzzy scheme

u=

  1 T − fˆ(x|Θ f ) + y¯ (n) m + k e , gˆ (x|Θg )

(32)

where f (x|Θ f ) and gˆ (x|Θg ) are governed by fˆ(x|Θ f ) gˆ (x|Θg )

= Θˆ Tf ξ(x) = Θˆ Tg η(x)

(33) (34)

. In addition, to enhance the adaptability of the indirect fuzzy logic mechanism, adaptive laws are employed to update the true values of the free parameters. The updated laws can be designed as: Θ˙ˆ f Θˆ˙

g

= −Υ1 Ξξ(x)

(35)

= −Υ2 Ξη(x)u

(36)

, where Υ1 and Υ2 are positive design parameters used to determine the control performance, η(x) is the vector for approaching g(·), and Ξ represents the additional terms in the controllers, which are formulated during the control design process. 5. Synthesis of the three control subsystems 5.1. θ-subsystem In practice, maintaining the upright position of the vehicle body is crucial for the WIP vehicle because after the pendulum-like vehicle body falls down, the vehicle cannot be recovered. Furthermore, it is a challenge to maintain the upright position in an unstable equilibrium for subsystem (18), so more control effects are needed to sustain the stability of the θ-subsystem. Therefore, a variable structure control method with adaptive laws is required to deal with this special underactuated subsystem. Note that θ can be used directly to represent the tracking error of the tilt angle of the vehicle body since the desired tilt angle of the pendulum-like vehicle body is θd = 0. In addition, θ-subsystem and υ-subsystem share the same control input τυ , so to clarify the explicit relationship of every variable, this control input can be calculated by (17), before substituting τυ into (18). As a result, an underactuated θ-subsystem without control effects can be obtained as follows. m p l2 2 m p gl m p l cos θ θ¨ = ω sin 2θ + sin θ − υ˙ (37) 2Jθ Jθ Jθ The longitudinal acceleration υ˙ from (37) can be taken as a virtual input, which allows the original underactuated dynamics to be converted into actuated dynamics. In particular, to decrease the dependency on the accurate dynamic model, the indirect fuzzy logic mechanism is employed to support the sliding mode control approach

9

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when constructing the controller, where the sliding mode control enhances the robustness to parameter variation and external disturbance, while the fuzzy logic mechanism provides adaptability to the dynamic environment [41, 42]. The sliding mode manifold is defined as s = θ˙ + cθ, (38) where c is a positive design parameter. According to the fuzzy logic mechanism, the estimated entries, i.e., fˆθ (x|Θ f θ ) and gˆ θ (x|Θgθ ), are designed to make the controller. The optimal parameters are assumed to be   ∗ ˆ Θ f θ = arg min sup | fθ (x|Θ f θ ) − fθ (x)| (39) Θ f θ ∈Ω f θ x∈Rn   (40) Θ∗gθ = arg min sup |ˆgθ (x|Θgθ ) − gθ (x)| , Θgθ ∈Ωgθ x∈Rn

where Ω f θ and Ωgθ are the sets for Θ f θ and Θgθ , respectively. Furthermore, the minimum approximate errors can be defined as follows. fˆθ (x|Θ∗f θ ) − fθ (x)

fθ

=

gθ

= gˆ θ (x|Θ∗gθ ) − gθ (x)

(41) (42)

Define Θ˜ f θ = Θˆ f θ − Θ∗f θ . Due to the universal approximation theorem of fuzzy system, we conclude that  f θ and gθ can converge to a compact set, and thus it follows that Θ˙˜ f θ = Θ˙ˆ f θ . Therefore, the adaptive laws can be introduced to update the true values of the controllers, which enhances the adaptive capability of the closed-loop system. Theorem 5.1. For the θ-subsystem (18), if the sliding mode surface is selected as in (38), then all of the signals in the closed-loop θ-subsystem are uniformly ultimately bounded, and the tracking errors will converge to a compact set, with the indirect adaptive fuzzy logic-based sliding mode controllers as follows: 1





−Θˆ Tfθ ξ(x) + cθ˙ − k1 sgn(s) − k2 s

uθ

=

Θ˙ˆ f θ Θˆ˙

=

γ1 sξ(x)

(44)

=

γ2 sη(x)uθ

(45)

gθ

Θˆ Tgθ η(x)

(43)

, where sgn(·) is a signum function, and γ1 , γ2 are positive design parameters. Θˆ f θ and Θˆ gθ are the weight values, and ξ(x) and η(x) are their membership functions, respectively. Proof. Consider a positive Lyapunov candidate, as follows. V2 =

1 2 1 1 (Θ f θ − Θ∗f θ )T (Θ f θ − Θ∗f θ ) + (Θgθ − Θ∗gθ )T (Θgθ − Θ∗gθ ) s + 2 2γ1 2γ2

(46)

By differentiating V2 along the trajectory (37) with respect to time, we have 1 1 V˙ 2 = s s˙ + Θ˜ Tfθ Θ˙ˆ f θ + Θ˜ Tgθ Θ˙ˆ gθ γ1 γ2   1 1 ∗T ˙ = s Θ∗T Θ˜ T Θ˙ˆ f θ + Θ˜ Tgθ Θ˙ˆ gθ f θ ξ(x) +  f θ + Θgθ η(x)uθ + gθ uθ − λθ + γ1 f θ γ2 .   1 1 = s (Θˆ f θ − Θ˜ f θ )T ξ(x) + (Θˆ gθ − Θ˜ gθ )T η(x)uθ +  f θ + gθ uθ − λθ˙ + Θ˜ Tfθ Θ˙ˆ f θ + Θ˜ Tgθ Θ˙ˆ gθ γ1 γ2   1 1 ˙ T T T T T = s Θˆ f θ ξ(x) + Θˆ gθ η(x)uθ − λθ˙ +  f θ + gθ uθ − sΘ˜ f θ ξ(x) − Θ˜ gθ η(x)uθ + Θ˜ f θ Θˆ f θ + Θ˜ Tgθ Θ˙ˆ gθ γ1 γ2

(47)

After substituting (43)–(45) into (47), it holds that V˙ 2 = −k1 |s| − k2 s2 + s f θ + sgθ uθ

(48)

. According to the universal approximation theorem,  f θ → 0 and gθ → 0 as t → ∞, and thus V2 ≤ 0. Therefore, the theorem can be proved by the Lyapunov stability theory and LaSalle’s invariance theorem. 10

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5.2. υ-subsystem For the υ-subsystem, the control objective is to find an appropriate control input uυ such that the vehicle’s longitudinal velocity can track the desired value. It should be mentioned that the control input uθ , which is employed to maintain the upright position of θ-subsystem, can be considered as a disturbance to the control τυ . We define the tracking errors as eυ = υ − υc and eω = ω − ωc , where υc and ωc are the desired velocities generated by the trajectory planner. From (16)–(17), the error dynamics are formulated as follows. e˙ υ =

 1 Jθ Jθ m p lθ˙ 2 sin θ − m2p l3 (eω + ωc )2 sin θ cos2 θ − m2p l2 g sin θ cos θ − υ˙ c + uυ Ω Ωr

(49)

Similarly, fˆυ (x|Θ f υ ) and gˆ υ (x|Θgυ ) can be used to make the system controller, and the optimal parameters can be hypothesized as   ∗ ˆ (50) Θ f υ = arg min sup | fυ (x|Θ f υ ) − fυ (x)| Θ f υ ∈Ω f υ x∈Rn   Θ∗gυ = arg min sup |ˆgυ (x|Θgυ ) − gυ (x)| , (51) Θgυ ∈Ωgυ x∈Rn

where Ω f υ , Ωgυ are the sets for Θ f υ and Θgυ , respectively. Moreover, the minimum approximate errors can be governed by fυ

=

gυ

=

fˆυ (x|Θ∗f υ ) − fυ (x)

(52)

gˆ υ (x|Θ∗gυ ) − gυ (x)

(53)

.

Define Θ˜ f υ = Θˆ f υ − Θ∗f υ . By the universal approximation theorem, we conclude that  f υ and gυ can converge to a compact set, and thus it holds that Θ˙˜ f υ = Θ˙ˆ f υ . By using the adaptive laws to update the true values employed in the controllers, a theorem can be stated as follows. Theorem 5.2. For the υ-subsystem (17), the signals in the closed-loop υ-subsystem are uniformly ultimately bounded and the tracking errors converge to a compact set, with the indirect adaptive fuzzy logic-based controllers as follows: uυ Θ˙ˆ f υ Θˆ˙ gυ

=



1 Θˆ Tgυ η(x)

Θˆ Tfυ ξ(x) − k3 eυ



(54)

= γ3 eυ ξ(x)

(55)

= γ4 eυ η(x)uυ

(56)

, where k3 , γ3 , γ4 are positive design parameters. Θˆ f υ and Θˆ gυ are the weight values, and ξ(x) and η(x) are their membership functions, respectively. Proof. Consider a positive Lyapunov candidate as follows. V3 =

1 2 1 1 eυ + (Θ f υ − Θ∗f υ )T (Θ f υ − Θ∗f υ ) + (Θgυ − Θ∗gυ )T (Θgυ − Θ∗gυ ) 2 2γ3 2γ4

(57)

Differentiating V3 along the trajectory (17) with respect to time yields 1 1 V˙ 3 = eυ e˙ υ + Θ˜ Tfυ Θ˙ˆ f υ + Θ˜ Tgυ Θ˙ˆ gυ γ3 γ4 1 1 = eυ ( fυ + gυ uυ ) + Θ˜ Tfυ Θ˙ˆ f υ + Θ˜ Tgυ Θ˙ˆ gυ γ3 γ4 1 ˜ T ˙ˆ 1 Θ Θ f υ + Θ˜ Tgω Θ˙ˆ gω γ3 f υ γ4   1 1 = eυ (Θˆ f υ − Θ˜ f υ )T ξ(x) +  f υ + (Θˆ gυ − Θ˜ gυ )T η(x)uυ + gυ uυ + Θ˜ Tfυ Θ˙ˆ f υ + Θ˜ Tgυ Θ˙ˆ gυ γ3 γ4   = eυ Θˆ Tfυ ξ(x) + Θˆ Tgυ η(x)uυ +  f υ + gυ uυ − eυ Θ˜ Tfυ ξ(x) − eυ Θˆ Tgυ η(x)uυ ∗T = eυ (Θ∗T f υ ξ(x) +  f υ + Θgυ η(x)uυ + gυ uυ ) +

+

1 ˜ T ˙ˆ 1 Θ Θ f υ + Θ˜ Tgυ Θ˙ˆ gυ . γ3 f υ γ4 11

(58)

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After substituting (54)–(56) into (58), it follows that V˙ 3 = −k3 e2υ + eυ  f υ + eυ gυ uυ

(59)

. Obviously, V3 ≤ 0 since  f υ → 0 and gυ → 0 as t → ∞. Hence, the theorem can be proved by the Lyapunov stability theory and LaSalle’s invariance theorem. Using the previous results, we note that uθ and uυ are not the real control inputs for the system, so a composite controller for the WIP vehicle can be obtained as follows. τυ = uθ + uυ

(60)

Remark 2. The control input (60) is a composite controller where the control input uθ , which is the acceleration required to maintain the balance of the vehicle body, is taken as a type of external dynamic disturbance with a high frequency for the υ-subsystem. Since uθ and uυ can guarantee the stability of the respective subsystem, then τυ can ensure the stability of the entire underactuated system. 5.3. ω-subsystem The control objective for ω-subsystem is to find a control input τω such that the rotational tracking error, i.e., eω = ω − ωc , should converge to a compact set. For this purpose, the error dynamics of the ω-subsystem are formulated as m p l2 θ˙ sin 2θ d (eω + ωc ) − ω τω ˙c + (61) e˙ ω = − Jϕ + m p l2 sin2 θ r(Jϕ + m p l2 sin2 θ) . Likewise, fˆω (x|Θ f ω ) and gˆ ω (x|Θgω ) are used to design the system controller, and the optimal parameters can be hypothesized as   (62) Θ∗f ω = arg min sup | fˆω (x|Θ f ω ) − fω (x)| Θ f ω ∈Ω f ω x∈Rn   Θ∗gω = arg min sup |ˆgω (x|Θgω ) − gω (x)| , (63) Θgω ∈Ωgω x∈Rn

where Ω f ω , Ωgω are the sets for Θ f ω and Θgω , respectively. Similarly, the minimum approximate errors can be governed by fˆω (x|Θ∗f ω ) − fω (x)

fω

=

gω

= gˆ ω (x|Θ∗gω ) − gω (x).

(64) (65)

Define Θ˜ f ω = Θˆ f ω − Θ∗f ω . With the universal approximation theorem, we find that  f ω and gω can converge to a compact set, and thus we obtain Θ˙˜ f ω = Θ˙ˆ f ω . By introducing the adaptive laws to update the true values in the controllers, a theorem can be stated as follows. Theorem 5.3. For the ω-subsystem (16), all of the signals in the closed-loop ω-subsystem are uniformly ultimately bounded and the tracking errors converge to a compact set, with the indirect adaptive fuzzy logic-based controllers as follows: τω Θ˙ˆ f ω Θˆ˙ gω

=



1 Θˆ Tgω η(x)

Θˆ Tfω ξ(x) − k4 eω



(66)

= γ5 eω ξ(x)

(67)

= γ6 eω η(x)τω

(68)

, where k4 , γ5 , γ6 are positive design parameters, Θˆ f ω and Θˆ gω are the weight values, and ξ(x) and η(x) are their membership functions, respectively. Proof. Consider a positive Lyapunov candidate as follows. V4 =

1 2 1 1 e + (Θ f ω − Θ∗f ω )T (Θ f ω − Θ∗f ω ) + (Θgω − Θ∗gω )T (Θgω − Θ∗gω ) 2 ω 2γ5 2γ6

12

(69)

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By differentiating V4 along the trajectory (16) with respect to time, it holds that 1 1 V˙ 4 = eω e˙ ω + Θ˜ Tfω Θ˙ˆ f ω + Θ˜ Tgω Θ˙ˆ gω γ5 γ6 1 1 = eω ( fω + gω τω ) + Θ˜ Tfω Θ˙ˆ f ω + Θ˜ Tgω Θ˙ˆ gω γ5 γ6 1 ˜ T ˙ˆ 1 Θ Θ f ω + Θ˜ Tgω Θ˙ˆ gω γ5 f ω γ6   1 1 = eω (Θˆ f ω − Θ˜ f ω )T ξ(x) +  f ω + (Θˆ gω − Θ˜ gω )T η(x)τω + gω τω + Θ˜ Tfω Θ˙ˆ f ω + Θ˜ Tgω Θ˙ˆ gω γ5 γ6   = eω Θˆ T ξ(x) + Θˆ Tgω η(x)τω +  f ω + gω τω − eω Θ˜ T ξ(x) − eω Θˆ Tgω η(x)τω ∗T = eω (Θ∗T f ω ξ(x) +  f ω + Θgω η(x)τω + gω τω ) +

fω

(70)

fω

1 1 + Θ˜ Tfω Θ˙ˆ f ω + Θ˜ Tgω Θ˙ˆ gω . γ5 γ6 After substituting (66)–(68) into (70), it follows that V˙ 4 = −k4 e2ω + eω  f ω + eω gω τω

(71)

. Obviously, V4 ≤ 0 as  f ω → 0 and gω → 0 as t → ∞. Therefore, the theorem can be proved by the Lyapunov stability theory and LaSalle’s invariance theorem. Remark 3. Singularities exist in the controllers for the indirect adaptive control scheme, and this is an unavoidable and troublesome issue for adaptive control. This problem can be addressed by rewriting the scalar ΘT 1η(x) , (i = gi

θ, υ, ω), with the following terms as

ΘTgi η(x) , 2 ΘTgi η(x) + 



(72)

where  > 0 is a sufficiently small coefficient to avoid singularity. 6. Simulations In this section, we present the results of numerical simulations performed for a WIP vehicle to verify the feasibility and effectiveness of the proposed control strategy in the MATLAB environment. The physical parameters of the WIP vehicle were set as follows: m p = 3kg, mc = 5kg, mw = 1kg, d = 1.0m, r = 0.5m, l = 1.0m, Jw = 1.5kgm2 , Jc = 5kgm2 , and Jϕ = 1.5kgm2 . In addition, if the desired trajectory is assumed to be a circular path with a radius of R = 2m, then it can be described in Cartesian coordinates as follows. ⎧ ⎪ x = R cos ωd t ⎪ ⎪ ⎨ d yd = R sin ωd t ⎪ ⎪ ⎪ ⎩ ϕ =ω t d

d

It should be noted that the longitudinal and rotational velocities used in the proposed controllers can be computed by υd =

xd2 + y2d and ωd = ϕ˙ d , respectively. The vehicle’s initial conditions for the posture error vectors,

velocity vector, and posture state variables were taken as q¯ e (0) = [xe (0) ye (0) ϕe (0)]T = [−0.05 0 π/24]T , z˙(0) = T ˙ = [0 0 0]T , and z(0) = [xυ (0) ϕ(0) θ(0)]T = [0 0.01 0.15]T , respectively. For the indirect [υ(0) ω(0) θ(0)] adaptive fuzzy system, two fuzzy controllers were employed to control the longitudinal and rotational subsystems separately. The fuzzy membership functions ξ(xi ) and η(xi ), as shown in Fig.4, were set as follows.   μN M (xi ) = exp −((xi + π/6)/(π/24))2   μNS (xi ) = exp −((xi + π/12)/(π/24))2   μZ (xi ) = exp −(xi /(π/24))2   μPS (xi ) = exp −((xi − π/12)/(π/24))2   μPM (xi ) = exp −((xi − π/6)/(π/24))2 To satisfy the control performance requirements, the design coefficients of the proposed control scheme were set as follows: c = 2, k1 = 4, k2 = 2, k3 = 2, k4 = 1; σ1 = σ3 = 2, σ2 = 1; γ1 = γ3 = 10, γ2 = γ4 = 1;  = 13

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1

Membership function degree

0.9 0.8

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 -0.6

-0.4

-0.2

0 xi

0.2

0.4

0.6

Figure 4. Membership function for the proposed fuzzy logic.

0.001. The sign functions in the sliding mode control schemes were replaced by hyperbolic tangent functions, i.e., tanh(·), to reduce the inherent chattering caused by the variable structure of the control system. Using these design parameters, the practical and desired tracking trajectories on the x − y plane are plotted in Fig.5, while the tracking errors are shown in Fig.6. These results demonstrate that the posture of the WIP vehicle can track the reference trajectory very well using the proposed control methods. However, it should be noted that a dramatic regulation process occurred at the start point, which was because the underactuated vehicle body had to maintain an upright position while the vehicle’s posture converged to the desired trajectories. In addition, the enlarged inset in Fig.6 clearly demonstrates that the tracking errors xe , ye and ϕe converged to zero within 15 s, thereby indicating the this dramatic regulation process could be suppressed rapidly. 2

0.4

xd,yd x,y

1.5 1

0

xe / y e / φ e

y

0.5 Start point

-0.5

0.05

0.2

0

0.1

-0.05 4

6

8

10

12

30

35

0

-0.1

-1

-0.2

-1.5 -2

0.3

xe ye φe

-2 -1.5

-1

-0.5

0

x

0.5

1

1.5

2

2.5

Figure 5. Practical and desired tracking trajectories.

-0.3 0

5

10

15

20

25 t (s)

40

45

50

Figure 6. Posture tracking errors for the WIP vehicle.

The time responses of the velocity tracking results are shown in Fig.7, which indicate the effects of longitudinal and rotational velocity tracking. These results demonstrate that the practical velocity variables υ and ω can track the desired velocities, υc and ωc , generated by the trajectory planner within finite time, thereby verifying the effectiveness of the indirect adaptive fuzzy controllers. In addition, it should be noted that an abrupt change occurred at about 2.5 s. This phenomenon indicates that control of the longitudinal velocity is more challenging because the longitudinal velocity controlled by the torque uυ but it is also affected by the disturbance-like virtual control input uθ . In particular, the stabilities of the tilt angle and its angular velocity are of crucial importance for control purposes. Their time responses are shown in Fig.8, which indicate that θ and θ˙ can converge rapidly to zero. Thus, these results verified that the composite controller produced by combining indirect adaptive fuzzy control and the sliding mode technique achieved acceptable control performance for the underactuated θ-subsystem. Figures 9 and 10 show the estimated processes for the fuzzy logic system. The simulation results demonstrate 14

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6 υ , υc

1

2

0

0 -2

υ υr

2

4

-1

0

5

0

2

4

10

15

20

6

25

8

30

35

40

45

ω, ωc

0.4

ω ωr

0.3

0.3 0.2

0.25 0.2

0.1

0.15

0

5

10

50

2

4

6

15

20

25 t (s)

8

30

10

35

40

45

50

Figure 7. Response of velocity tracking over time.

0.4

θ

0.3 0 0.2 0 0.1

0

0

5

10

15

20

5

10

15

20

25

30

35

40

45

50

25 30 t (s)

35

40

45

50

0.2

T

0.1 0

-0.1 -0.2 0

Figure 8. Responses of the tilt angle and angular velocity.

that the proposed controllers exhibited better adaptive features with the designed closed-loop system because of the indirect fuzzy logic mechanism. Our further analysis showed that fˆυ (x|Θ f υ ), gˆ υ (x|Θgυ ), and fˆω (x|Θ f ω ) could converge to their respective values within a compact set in 5 s despite the presence of elastic oscillations at the initial time. By contrast, the time response of gˆ ω (x|Θgω ) did not agree well with the true value of gω (x|Θgω ), but its tracking error converged to a compact set, which implies that the proposed adaptive laws are effective and that the global stability is also guaranteed. 7. Conclusion In this study, we proposed an error data-based trajectory planner and indirect adaptive fuzzy control for a WIP vehicle with an underactuated suspension, which is subject to a nonholonomic constraint. Using the closed-loop planner, the nonholonomic constraint and underactuated problems can be resolved on kinematic and dynamic levels, respectively. To address the underactuated problem of the tilt angle dynamics, we treat the longitudinal accelerated velocity as a virtual control input to ensure that the inverted pendulum-like vehicle body maintains an upright position, which is an unstable equilibrium for the system. The proposed control method is a type of composite control that synthesizes indirect fuzzy control and the variable structure technique in an effective manner. Using an indirect fuzzy mechanism, the controller can merge the successful experiences of operators. By employing the sliding mode control technique, the controller is robust to uncertain disturbance. In addition, adaptive laws are employed to ensure independence from the vehicle mechanical parameters in the controller. These advantages are crucial and significant for the performance of the WIP vehicle control system. It should 15

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0.3

fυ fˆX

0

16

fω fˆZ

0.2

fZ , fˆZ

0.1

fX , fˆX

-20 -40 0

5

10

15

20

25

30

35

40

45

5

-10

0

5

10

15

20

25 30 t (s)

35

40

45

5

10

15

20

25

30

35

40

45

50

Figure 9. Responses of fυ , fˆυ and gυ , gˆ υ

50

gω gˆ Z

1

gˆX

-5

0

1.5

gZ, gˆZ

gX , gˆX

-0.1

gυ

0

-15

50

0

0.5 0

-0.5 0

5

10

15

20

25 30 t (s)

35

40

45

50

Figure 10. Responses of fω , fˆω and gω , gˆ ω

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